The mood of your fans improves as your team wins matches and scores lots of goals. However a 3-0 defeat at home isn't popular at all. The fans are fickle; one week you're a king and a hero, next week they're furious with you! The heart of your supporters club is made up of a hard-core of devoted fans. They won't abandon their team just for a few minor setbacks, but the size of the fan club changes in relation to club fortunes and misfortunes. A die-hard fan club is built up over many seasons, so the top teams who have been in the highest division for several seasons generally have the largest supporters clubs.
If you value your fans, try not to lose by a large margin in home matches. The effect is much worse if your team doesn't score any goals while the other team scores plenty. When this happens the fans become very upset. Results away from home still have an effect, but it is not so pronounced. Conversely, scoring lots of goals, and winning at home is the best way to increase the size of your fanclub. A win in general doesn't bring in many fans. Scoring a lot of goals and winning is far better than winning 1-0 so try and play your matches accordingly.
The amount of fans will change on the daily update immediately after your home countries matchday (competition or friendly). The number of fans joining your club depends on their mood (better mood leads to more fans) and on fan club size with respect to division level. Each division has a soft cap on maximum size of team's fan club. The closer you are to the soft cap, the harder it becomes to get new fans.
Remember that when you lose a cup game you will take a fair hit fan wise, so don't be upset when it happens as it is expected.
Every new team starts with 100 fans, but this (at least initially!) increases quickly.